Answer :
To expand the expression [tex]\((2x - 1)^6\)[/tex], we can use the Binomial Theorem. The Binomial Theorem states that:
[tex]\[
(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k
\][/tex]
In this case, [tex]\(a = 2x\)[/tex], [tex]\(b = -1\)[/tex], and [tex]\(n = 6\)[/tex]. Let's expand [tex]\((2x - 1)^6\)[/tex] step-by-step:
1. Identify Each Term: We'll find each term using the formula [tex]\(\binom{n}{k} \cdot (2x)^{n-k} \cdot (-1)^k\)[/tex].
2. Calculate the Terms:
- For [tex]\(k = 0\)[/tex]: [tex]\(\binom{6}{0} \cdot (2x)^6 \cdot (-1)^0 = 1 \cdot 64x^6 \cdot 1 = 64x^6\)[/tex]
- For [tex]\(k = 1\)[/tex]: [tex]\(\binom{6}{1} \cdot (2x)^5 \cdot (-1)^1 = 6 \cdot 32x^5 \cdot (-1) = -192x^5\)[/tex]
- For [tex]\(k = 2\)[/tex]: [tex]\(\binom{6}{2} \cdot (2x)^4 \cdot (-1)^2 = 15 \cdot 16x^4 \cdot 1 = 240x^4\)[/tex]
- For [tex]\(k = 3\)[/tex]: [tex]\(\binom{6}{3} \cdot (2x)^3 \cdot (-1)^3 = 20 \cdot 8x^3 \cdot (-1) = -160x^3\)[/tex]
- For [tex]\(k = 4\)[/tex]: [tex]\(\binom{6}{4} \cdot (2x)^2 \cdot (-1)^4 = 15 \cdot 4x^2 \cdot 1 = 60x^2\)[/tex]
- For [tex]\(k = 5\)[/tex]: [tex]\(\binom{6}{5} \cdot (2x)^1 \cdot (-1)^5 = 6 \cdot 2x \cdot (-1) = -12x\)[/tex]
- For [tex]\(k = 6\)[/tex]: [tex]\(\binom{6}{6} \cdot (2x)^0 \cdot (-1)^6 = 1 \cdot 1 \cdot 1 = 1\)[/tex]
3. Combine the Terms: Adding up all these terms gives us the expanded form:
[tex]\[
64x^6 - 192x^5 + 240x^4 - 160x^3 + 60x^2 - 12x + 1
\][/tex]
Therefore, the correct expanded form of [tex]\((2x - 1)^6\)[/tex] is:
[tex]\[
64x^6 - 192x^5 + 240x^4 - 160x^3 + 60x^2 - 12x + 1
\][/tex]
This matches option D.
[tex]\[
(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k
\][/tex]
In this case, [tex]\(a = 2x\)[/tex], [tex]\(b = -1\)[/tex], and [tex]\(n = 6\)[/tex]. Let's expand [tex]\((2x - 1)^6\)[/tex] step-by-step:
1. Identify Each Term: We'll find each term using the formula [tex]\(\binom{n}{k} \cdot (2x)^{n-k} \cdot (-1)^k\)[/tex].
2. Calculate the Terms:
- For [tex]\(k = 0\)[/tex]: [tex]\(\binom{6}{0} \cdot (2x)^6 \cdot (-1)^0 = 1 \cdot 64x^6 \cdot 1 = 64x^6\)[/tex]
- For [tex]\(k = 1\)[/tex]: [tex]\(\binom{6}{1} \cdot (2x)^5 \cdot (-1)^1 = 6 \cdot 32x^5 \cdot (-1) = -192x^5\)[/tex]
- For [tex]\(k = 2\)[/tex]: [tex]\(\binom{6}{2} \cdot (2x)^4 \cdot (-1)^2 = 15 \cdot 16x^4 \cdot 1 = 240x^4\)[/tex]
- For [tex]\(k = 3\)[/tex]: [tex]\(\binom{6}{3} \cdot (2x)^3 \cdot (-1)^3 = 20 \cdot 8x^3 \cdot (-1) = -160x^3\)[/tex]
- For [tex]\(k = 4\)[/tex]: [tex]\(\binom{6}{4} \cdot (2x)^2 \cdot (-1)^4 = 15 \cdot 4x^2 \cdot 1 = 60x^2\)[/tex]
- For [tex]\(k = 5\)[/tex]: [tex]\(\binom{6}{5} \cdot (2x)^1 \cdot (-1)^5 = 6 \cdot 2x \cdot (-1) = -12x\)[/tex]
- For [tex]\(k = 6\)[/tex]: [tex]\(\binom{6}{6} \cdot (2x)^0 \cdot (-1)^6 = 1 \cdot 1 \cdot 1 = 1\)[/tex]
3. Combine the Terms: Adding up all these terms gives us the expanded form:
[tex]\[
64x^6 - 192x^5 + 240x^4 - 160x^3 + 60x^2 - 12x + 1
\][/tex]
Therefore, the correct expanded form of [tex]\((2x - 1)^6\)[/tex] is:
[tex]\[
64x^6 - 192x^5 + 240x^4 - 160x^3 + 60x^2 - 12x + 1
\][/tex]
This matches option D.
